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A function (in black) is convex if and only if the region above its graph (in green) is a convex set. A graph of the bivariate convex function x 2 + xy + y 2. Convex vs. Not convex
Convex analysis includes not only the study of convex subsets of Euclidean spaces but also the study of convex functions on abstract spaces. Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex function - a function in which the line segment between any two points on the graph of the function lies above the graph. Closed convex function - a convex function all of whose sublevel sets are closed sets. Proper convex function - a convex function whose effective domain is nonempty and it never attains minus infinity. Concave ...
In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function , and one can freely convert between the two, but the characteristic function as defined below is better ...
A convex optimization problem is defined by two ingredients: [5] [6] The objective function, which is a real-valued convex function of n variables, :;; The feasible set, which is a convex subset.
For every proper convex function : [,], there exist some and such that ()for every .. The sum of two proper convex functions is convex, but not necessarily proper. [4] For instance if the sets and are non-empty convex sets in the vector space, then the characteristic functions and are proper convex functions, but if = then + is identically equal to +.
In complex geometry and analysis, the notion of convexity and its generalizations play an important role in understanding function behavior. Examples of classes of functions with a rich structure are, in addition to the convex functions, the subharmonic functions and the plurisubharmonic functions.
In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.